>No, I'm trying from IE right now. I'll try from Opera then. But I do not see a way to use Contacts from this e-mail anyway. There is also a button in the right top corner of the new e-mail I'm trying to write, but I do not understand, what does it do. It doesn't have any tooltips and clicking on it also does nothing.

I think you're a victim of a pop-up blocker... All those buttons open a new window; and if they don't then there's something not allowing that window to open.